Quartz resonator pressure transducers and methods of operation

ABSTRACT

A cylindrical quartz crystal transducer that exhibits a low probability of twinning, and uses a combination of resonator signal inputs at the B-mode and C-mode frequencies to calculate resonator temperature. Crystallographic orientations are disclosed where combinations of B-mode and C-mode resonant frequencies exist that are sufficiently independent of pressure to enable accurate calculation of temperature under transient conditions. Such a transducer is usable at higher pressures and temperatures than conventional quartz pressure transducers. Furthermore, because the structure allows a choice of crystallographic orientation, other characteristics of the transducer, such as increased pressure sensitivity and activity dip-free operation, may be optimized by varying crystallographic orientation.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a divisional of U.S. patent application Ser. No.13/839,238, filed Mar. 15, 2013, which will issue as U.S. Pat. No.9,528,896 on Dec. 27, 2016, the disclosure of which is herebyincorporated herein in its entirety by this reference.

FIELD

The present disclosure relates generally to apparatuses and methods formeasurement of high pressures under extreme and transient temperatureconditions and, more particularly, to quartz resonator pressuretransducers configured to provide integral temperature compensation andmethods of using same.

BACKGROUND

Quartz resonator pressure transducers have been used successfully in thedownhole environment of oil and gas wells for several decades and arestill the most accurate means of determining bottom-hole pressure. Whilemany measurements of these downhole pressures are made under static orslowly varying pressure and temperature conditions, some significantsituations, however, require pressure measurements under transientconditions where either or both of the temperature and pressure arechanging. The range of static to dynamic measurement conditions, theeconomic drive for less expensive devices, and the increasing levels ofpressures and temperatures arising as the world oil and gas explorationand production industry drills deeper and deeper, have spurredcontinuing developments in quartz resonator pressure transducers.

The first commercially successful quartz resonator pressure transducer,as disclosed in U.S. Pat. Nos. 3,561,832 and 3,617,780, the disclosureof each of which is hereby incorporated herein in its entirety by thisreference, was introduced by Hewlett Packard (“HP”) in the 1970's. Thistransducer was of a cylindrical design with the resonator formed in aunitary fashion in a single piece of quartz. End caps of quartz wereattached to close the structure. FIG. 1A shows this configuration, whichcontains resonator 1 a unitary (integral) with body 2 a, two end caps 3a, and two glass joints 4 a. This device was relatively large,approximately 1 inch diameter and 4 inches long. The unitary body andresonator are expensive to manufacture. Also, two major disadvantageswere caused by the large size. Large stress distributions occurthroughout the structure under transient conditions because thetemperature distribution is slow to equilibrate. These stresses causeerrors in the pressure measurement. Also, it is not practical to obtaina temperature measurement, necessary for temperature compensation, closeto the actual location of the pressure measurement, e.g., the resonator,because of the large transducer size. This lack of proximity results intemperature errors in transient conditions because the temperature at atemperature transducer used to temperature compensation may not be thesame as the required temperature located at the resonator itself. Bothof these problems restricted the use of this concept to the more benign,nearly static cases.

A somewhat smaller size transducer was introduced in the 1980's byQuartztronics, Inc., of Murray, Utah, and commercialized by HalliburtonCompany through its Halliburton Services operating unit, now part ofHalliburton Energy Services. This device, as described in U.S. Pat. Nos.4,550,610 and 4,660,420, the disclosure of each of which is herebyincorporated herein in its entirety by this reference, was similar tothe unitary HP design, except diametrically opposed flats were added tothe cylindrical shape to create a non-uniform stress distribution in theresonator under pressure. FIG. 1B shows this structure, which containsresonator 1 b unitary with body 2 b, two end caps 3 b, two glass joints4 b, and a pair of flats 5 b (backside flat not shown in FIG. 1B). Thesmaller size of the Quartztronics transducer reduced the cost, and theflats increased the pressure sensitivity while reducing the temperaturesensitivity. The smaller size also reduced the amount of undesiredstress distribution from non-uniform thermal distributions and enabledtemperature to be measured closer to the pressure measurement location.

Another quartz resonator transducer design was introduced in the 1990'sby Quartzdyne, Inc. of Murray, Utah. This device eliminated thebody/resonator unitary structure by simply bonding a convex-convexresonator between two end caps. FIG. 1C shows this configuration, whichcontains resonator 1 c, two end caps 3 c, and two glass joints 4 c.Besides low cost, the physical size was small enough to move thetemperature measurement point to within a few millimeters of thepressure measurement location.

The foregoing three quartz resonator transducers each use a singleresonant mode, the slow-shear thickness mode, or C-mode, to determinepressure external to the transducer. A temperature compensation signalis supplied with an independent temperature measurement device locatedas close as possible to the pressure measurement (resonator) location.

In light of recognition of a need for good pressure measurements intransient conditions, researchers have explored different ways to use adual-mode transducer, wherein two resonant modes are driven by thedriving circuits of the transducer at the same time. In a dual-modetransducer, one resonant mode is mainly dependent on pressure, the othermode is mainly dependent on temperature. This approach would provide atemperature measurement located exactly where the pressure measurementwas made, eliminating one important error source. One mode, usually theC-mode, is used to measure the pressure, and a second mode, thefast-shear mode, or B-mode, is used to determine the temperature. Withthe two unknowns, pressure and temperature, and two simultaneousmeasurements, one can solve the two equations. However, during atransient condition, the non-uniform stresses in the structure, arisingfrom non-uniform temperature distributions therein, changes the resonantfrequencies. If the B-mode is pressure sensitive, this frequency errorwill cause an error in the temperature calculation which will, in turn,cause an error in the calculation of the pressure. One is forced toperform a series of iterative calculations that may not lead to accuratepressure and temperature answers. The simplicity and accuracy of thepressure calculation in this case is greatly enhanced if the B-mode isnot pressure sensitive. This fact has driven research efforts indual-mode transducers to find B-modes with no pressure sensitivity whilestill having a C-mode available for the pressure measurement.

It has been recognized that one way to obtain a B-mode that isindependent of pressure is to change the crystallographic orientation ofthe quartz in the device. This approach led to the SBTC orientation, asdescribed in Michel Valdois, Bikash K. Sinha, and Jean Jacques Boy,EXPERIMENTAL VERIFICATION OF STRESS COMPENSATION IN THE SBTC-CUT, IEEETrans. Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 36, p.643, 1989, the disclosure of which is hereby incorporated herein in itsentirety by this reference. The shape of the SBTC orientation transduceris identical to that shown in FIG. 1A. Although this concept wassuccessful in obtaining a B-mode with no pressure sensitivity, theC-mode was not usable in a practical oscillator circuit because of highelectrical resistance.

Another approach to quartz resonator transducer design is described inU.S. Pat. No. 4,562,375, the disclosure of which is hereby incorporatedherein in its entirety by this reference. This structure uses aresonator bonded between two end caps, similar to the structure depictedin FIG. 1C. However, the resonator in this transducer design includesslots to isolate most of the perimeter of the resonator, leaving smallbridges to transfer the force from the endcaps along a specificdirection such that the B-mode will be pressure insensitive. Thisstructure has never been used commercially. The reason is believed to bethat the design cannot withstand the high pressures experienced in awellbore without failure because of the stress concentrations in thecorners of the slots.

To date, the only commercially successful dual-mode quartz pressuretransducer is the CQG (Crystal Quartz Gauge), offered by Schlumbergerand described in U.S. Pat. Nos. 4,547,691; 5,394,345 and 6,147,437, thedisclosure of each of which is hereby incorporated herein in itsentirety by this reference. It is a radical departure from the previousstructures in that, although the exterior is essentially cylindrical,the resonator is suspended across the inside diameter with the plane ofthe resonator extending in the axial direction. FIG. 2 shows the CQGstructure, which contains resonator 1 d unitary with body 2 d, two endcaps 3 d, and two glass joints 4 d. The drawings of the patents relatingto this structure show it with no flats, as well as with flats 5 d.There is an additional small flat 6 d, as shown. This small flat 6 d isshallow enough that it does not appreciably affect the stress magnitudeor distribution in the resonator 1 d, and is apparently used forassembly facilitation to crystallographically orient the end caps 3 dwith the body 2 d. Whereas all previous transducer structures previouslymentioned herein exhibit a two-dimensional stress in the resonator, theCQG structure has an almost uniaxial stress pattern in the resonator.The orientation of the resonator can be selected so that the B-mode ispressure insensitive.

There have been several attempts to accomplish dual-mode operation in astructure resembling the Quartztronics design employed by Halliburton,where the stress in the resonator is two-dimensional, but not uniform.One approach is described in U.S. Pat. No. 6,455,985, the disclosure ofwhich is hereby incorporated herein in its entirety by this reference.In this design, the unitary body with the resonator is cylindrical.However, the end caps, while being cylindrical on the outside, arestiffened inside along one direction to create a non-uniform stress inthe resonator. A second approach is described in U.S. Pat. No. 6,111,340(the “'340 patent”), the disclosure of which is hereby incorporatedherein in its entirety by this reference. In this design, the structureis the same as the shape employed in the Quartztronics/Halliburtontransducer, the only difference being that it is a dual-mode device.However, the '340 patent demonstrates that, even with very deep flatsthat take up two-thirds (⅔) of the wall thickness, it is not possible torender the B-mode completely independent of pressure but suggests that,if the pressure sensitivity of the B-mode can be reduced sufficientlywith the use of the flats, a usable device is possible. This inventionwould appear to be useful only if the flats are very deep. However, thestress concentrations associated with deep flats may lead to cracking ortwinning, and are not consistent with an ongoing desire prevalentthroughout the industry to extend the upper limits of pressure andtemperature measurement.

As disclosed in Schodowski, RESONATOR SELF-TEMPERATURE-SENSING USING ADUAL-HARMONIC-MODE CRYSTAL OSCILLATOR, 43r^(d) Annual Symposium onFrequency Control, 1989, p. 2 and U.S. Pat. No. 4,872,765 to Schodowskias well as in U.S. Pat. No. 4,545,638 to EerNisse and Ward, thedisclosure of each of which is hereby incorporated herein in itsentirety by this reference, temperature compensation is accomplished byusing two harmonically related resonances, typically the C-modefundamental and 3r^(d) overtone. The temperature is calculated using theformula 3*f_(Cfund)−f_(C3rd). This use of harmonically relatedvibrational modes must, however, include the fundamental mode to obtainthe temperature sensitivity. As the fundamental mode is more spread outthan the overtones, a device employing this approach requires arelatively large resonator bore diameter, leaving many unwanted modesnot clamped and increasing the chances for an activity dip.

One limitation common to all the quartz resonator pressure transducerconcepts is a tendency toward twinning at high applied stress andtemperature. Twinning is not reversible and renders the device unusable.In the past few years, the pressures and temperatures encountered in thedeeper wells have exceeded the capabilities of the CQG structure, whichhas stress concentrations in edges and corners. Twinning is lessprevalent in designs such as those of FIGS. 1A and 1C with uniformtwo-dimensional stress in the resonator. Also, because these cylindricalstructures minimize the number of corners and edges, cracking andtwinning in the rest of the structure is less probable.

BRIEF SUMMARY

Embodiments of the present disclosure employ a substantially cylindricalquartz crystal transducer structure that exhibits a low probability oftwinning, and uses a combination of signal inputs at the B-mode andC-mode frequencies to calculate temperature. A range of crystallographicorientations are available where a combination of the B-mode and C-modefrequencies exists that is sufficiently independent of pressure toenable accurate calculation of temperature under transient conditions.Thus, quartz structures, according to the present disclosure, may beused to provide a dual-mode pressure transducer with superiorperformance in comparison to conventional quartz pressure transducers.In addition, quartz structures of transducers of the present disclosureare less prone to twinning, so such transducers can be used at higherpressures and temperatures than conventional quartz pressuretransducers. Furthermore, because the structure of the presentdisclosure allows a choice of crystallographic orientation, the designeris free to optimize other characteristics, such as increased pressuresensitivity and activity dip-free operation, by varying crystallographicorientation.

In one embodiment, a dual-mode pressure transducer comprises a quartzcrystal structure having a crystallographic orientation with phi betweenabout 24° and less than about 30°, wherein the quartz crystal structurecomprises a substantially cylindrical body having a longitudinal bore;and a disc-shaped resonator carried by the body and extendingtransversely across the longitudinal bore.

In another embodiment, a dual-mode pressure transducer comprises aquartz crystal structure comprising a resonator and having acrystallographic orientation adapted to provide a combination of signalinputs from a non-fundamental B-mode resonant frequency and anon-fundamental C-mode resonant frequency of the resonator to enablecalculation of temperature of the resonator under transient conditions,wherein the quartz crystal structure comprises a substantiallycylindrical body having a longitudinal bore and the resonator isdisc-shaped, carried by the body and extends transversely across thelongitudinal bore.

In a further embodiment, a method of measuring a temperature-compensatedpressure using a quartz crystal structure comprises stimulating, undertransient temperature conditions, a resonator under external pressureapplied to the quartz crystal structure to provide signal inputs from anon-fundamental B-mode resonant frequency and a non-fundamental C-moderesonant frequency and using a combination of the signal inputs tocompensate a pressure determined from the non-fundamental C-moderesonant frequency signal input.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a perspective cutaway view of a prior art quartz pressuretransducer configuration;

FIG. 1B is a perspective cutaway view of another prior art quartzpressure transducer configuration;

FIG. 1C is a perspective cutaway view of a further prior art quartzpressure transducer configuration;

FIG. 2 is a perspective cutaway view of yet another prior art quartzpressure transducer configuration;

FIG. 3 is a graph of frequency shift versus temperature for phi=26° atvarious pressures;

FIG. 4 is a graph of slope of frequency shift versus temperature forphi=26° at various pressures;

FIG. 5 is a graph of calculated pressure sensitivity of the B- andC-modes of a round quartz pressure transducer versus phi angle;

FIG. 6 is a graph of the error factor in calculated static pressure forf_(C)+f_(B) temperature errors due to the H stress;

FIG. 7 is a graph of slope in Hz/psi an indicated combination of f_(C)and f_(B) versus T, calculated using experimental data from a phi=26°quartz pressure transducer; and

FIG. 8 is a schematic diagram of a circuit suitable for use withembodiments of a quartz crystal pressure transducer according toembodiments of the present disclosure.

DETAILED DESCRIPTION

To facilitate a more complete understanding of embodiments of thepresent disclosure and their operation, it is prudent here to develop abasis for evaluating what errors will occur in a dual-mode transducergiven a known level of pressure sensitivity of the B-mode, or in thepresent case, of the sum of f_(C) and f_(B). This will be done usingsome conventional methods for describing the relationship of f_(C) andf_(B) with pressure and temperature.

Equations for using two modes for computing pressure, P, andtemperature, T, are described, for instance, in R. J. Besson et. al., ADUAL-MODE THICKNESS-SHEAR QUARTZ PRESSURE SENSOR, IEEE Trans.Ultrasonics, ferroelectrics, and Frequency Control, Vol. 40, p. 584,1993, the disclosure of which is hereby incorporated herein in itsentirety by this reference. These equations express the pressure andtemperature in two-dimensional power series expansions in the twovariables, f_(C) and f_(B), which are the measured frequencies of theC-mode and B-mode, respectively. This approach works well because of thesmooth behavior of f_(C) and f_(B) with pressure and temperature.

FIG. 3 shows the behavior of f_(C) for a device comprising a transducerstructured according to an embodiment of the present disclosure whensubjected to a static P and T. The f_(C) value at atmospheric pressureand a temperature of 25° C. is typically 7.26 MHz. For the behaviorillustrated in FIG. 3, the outside diameter of an embodiment of thedevice configured as shown in FIG. 1C is 0.575 inch, the bore diameterof the end caps is 0.300 inch and bore depth is 0.120 inch. Theresonator is a 3^(rd) overtone blank with a diopter of 2.5 on bothsides. The crystallographic orientation is phi=26° and theta is near34°. As known to those of ordinary skill in the art, the angle phi isthe angle between the X-axis and the line of intersection of the blankor atomic plane with the XY-plane of a conventionally employedrectangular coordinate system, while theta is the angle between theZ-axis and the plane of the blank or atomic plane. The appropriate thetaangle may be chosen such that the first order temperature coefficient ofthe C-mode is zero. This may be calculated according to the followingequation, known to those of ordinary skill in the art:Θ=35.25°−(11/180)×ΦThe viable pressure range extends to about 30,000 psi and thetemperature range is 25° C. to 200° C.

Embodiments of the present disclosure may be physically implementedutilizing the quartz crystal structures illustrated herein in FIGS. 1Aand 1C. As noted previously, FIG. 1A depicts a unitary resonator andbody with end caps at each end of the body, whereas FIG. 1C depicts aresonator sandwiched between two end caps comprising the body. Forexample, quartz crystal structures in accordance with the presentdisclosure may include a convex-convex resonator and two end caps. Inother embodiments, the quartz crystal structures in accordance with thepresent disclosure may include other resonators configurations such asplano-plano and plano-convex.

Referring to FIG. 3, changes in f_(C) exhibit smooth behavior withshifts in pressure over a wide range of temperatures. The followingapproach is taken for a theoretical development of the possible errorsin using a dual-mode device in a transient situation. At a given staticpressure P₁ and static temperature T₁, the f_(C) and f_(B) behavior canbe described for small excursions in P and T around P₁ and T₁ with aTaylor series expansion. The expansion is limited to terms linear in Pand T and any cross-products of P and T are ignored.

$\begin{matrix}{f_{C} = {f_{C\; 1} + {\frac{\partial f_{C}}{\partial T}*\left( {T - T_{1}} \right)} + {\frac{\partial f_{C}}{\partial P}*\left( {P - P_{1}} \right)}}} & (1) \\{f_{B} = {f_{B\; 1} + {\frac{\partial f_{B}}{\partial T}*\left( {T - T_{1}} \right)} + {\frac{\partial f_{B}}{\partial P}*\left( {P - P_{1}} \right)}}} & (2)\end{matrix}$

If the constants C_(T), C_(P), B_(T), and B_(P) are defined as follows,

$\begin{matrix}{{C_{T} = {\frac{1}{f_{C\; 1}}\frac{\partial f_{C}}{\partial T}}}{C_{P} = {\frac{1}{f_{C\; 1}}\frac{\partial f_{C}}{\partial P}}}{B_{T} = {\frac{1}{f_{C\; 1}}\frac{\partial f_{B}}{\partial T}}}{{B_{P} = {\frac{1}{f_{C\; 1}}\frac{\partial f_{B}}{\partial P}}},}} & (3)\end{matrix}$then equations 1 and 2 can be written asf _(C) =f _(C1) +f _(C1) C _(T)*(T−T ₁)+f _(C1) C _(P)*(P−P ₁)  (4)f _(B) =f _(B1) +f _(B1) B _(T)*(T−T ₁)+f _(B1) B _(P)*(P−P ₁)  (5)

Equation 4 can be used in development of an error budget by answeringthe question: How accurate does one need to know T to calculate P to agiven accuracy level? Using Equation 4, we can solve for an error inf_(C), Δf_(C) , caused by an error in T, ΔT, given that P=P₁.Δf _(c) =f _(C1) C _(T) ΔT.  (6)Now, assuming that T=T₁, Equation 4 can be solved for P in terms off_(C).

$\begin{matrix}{{P - P_{1}} = {\frac{\left( {f_{C} - f_{C\; 1}} \right)}{f_{C\; 1}C_{P}}.}} & (7)\end{matrix}$If Equation 6 is substituted into Eq. 7, the error in P, ΔP, can beestimated as

$\begin{matrix}{{\Delta\; P} = {\frac{C_{T}}{C_{P}}\Delta\;{T.}}} & (8)\end{matrix}$As shown by the equation, a combination of a low temperature sensitivity(small C_(T)) and a large pressure sensitivity (large C_(P)) minimizesthe P error, ΔP, due to an error in T.

When there is a transient situation involving a temperature shift, thereare stresses created in the resonator due to a non-uniform temperaturedistribution in the resonator. This stress value at the center of theresonator causes a frequency shift that is an error in indicatedpressure, which will be termed H. It is conventional to use f_(B) forthe calculation of T. Using Equation 5, the error in T, ΔT, caused by His

$\begin{matrix}{{\Delta\; T} = {\frac{B_{P}}{B_{T}}{H.}}} & (9)\end{matrix}$

Thus, an error in the calculated pressure from ΔT caused by H isrepresented by

$\begin{matrix}{{\Delta\; T} = {\frac{C_{T}}{C_{P}}\frac{B_{P}}{B_{T}}{H.}}} & (10)\end{matrix}$This is the error that arises from the pressure sensitivity of f_(B). Itis known in the art to have B_(P) small and B_(T) large, as well assmall C_(T) and large C_(P). The numbers provided in U.S. Pat. No.6,111,340 may be used to calculate the coefficient in Equation 10 thatthe inventors therein considered practical, i.e., “substantiallyinsensitive” to pressure, (|B_(P)|≈|C_(P)|, and |C_(T)|≈3 ppm/°C. and|B_(T)|≈28 ppm/°C.). The coefficient is ≈0.107. This indicates that theerror in calculating P due to the non-uniform temperature distributionis approximately ten times (10×) less than H, the indicated error in Parising from the non-uniform temperature distribution during a transientevent. One may proceed from here assuming that 10× is an approximatethreshold for practical dual-mode performance.

FIG. 4 shows the slope of FIG. 3 in ppm/°C. for an embodiment of aquartz pressure transducer according to the present disclosure atphi=26°. The crystallographic orientation has been adjusted to minimizethe magnitude of the slope over the entire pressure and temperatureranges to be 3 ppm/°C. This number may be used as one design parameterfor evaluating an error budget and this number is approximately the sameover the range of phi from 22° to 30°. Also, the B_(T) coefficient isfound to be approximately 28 ppm/°C. over this phi range. One mayproceed with these two numbers assumed to be relatively constant overthe phi range under consideration herein for implementation of oneembodiment of the present disclosure.

One form of the present disclosure uses f_(C)+f_(B) for the temperaturecalculation. An equation may be derived for this case that is equivalentto Equation 10 for the error in P due to H. Assume that P=P₁. Then, thechange in temperature ΔT is calculated from Equations 4 and 5 by

$\begin{matrix}{{\Delta\; T} = {\frac{\left( {f_{C} + f_{B} - f_{C\; 1} - f_{B\; 1}} \right)}{\left( {{f_{C\; 1}C_{T}} + {f_{B\; 1}B_{T}}} \right)}.}} & (11)\end{matrix}$Assume that T=T₁. Then if the non-uniform stress is present and there isan error in the indicated pressure of H, the error in frequency forf_(C)+f_(B) is given by(f _(C) +f _(B) −f _(C1) −f _(B1))=(f _(c1) C _(P) +f _(B1) B_(P))H.  (12)When Equations 11 and 12 are combined, the error in T is given by

$\begin{matrix}{{\Delta\; T} = {\frac{\left( {{f_{C\; 1}C_{P}} + {f_{B\; 1}B_{P}}} \right)}{\left( {{f_{C\; 1}C_{T}} + {f_{B\; 1}B_{T}}} \right)}{H.}}} & (13)\end{matrix}$Equation 13 for the error in T, combined with Equation 8, provides uswith the equivalent of Equation 10:

$\begin{matrix}{{\Delta\; P} = {\frac{C_{T}}{C_{P}}\frac{\left( {{f_{C\; 1}C_{P}} + {f_{B\; 1}B_{P}}} \right)}{\left( {{f_{C\; 1}C_{T}} + {f_{B\; 1}B_{T}}} \right)}{H.}}} & (14)\end{matrix}$Equation 14 represents a significant aspect of the disclosure. Insteadof B_(P) and B_(T) in Equation 10, which are in ppm/psi and ppm/° C., inthe present disclosure the coefficients in the parenthesis arecalculated using Hz/psi and Hz/° C. The power of this approach becomesevident when looking at FIG. 5. There, the calculated values are shownfor f_(C1)*C_(P) and f_(B1)*B_(p), as well as the sum(f_(C1)*C_(P)+f_(B1)*B_(p)) for a round sensor design according to anembodiment of the disclosure over the range of phi angles from 22° to30°. Since the values are of opposite sign, the sum trends toward zerofor phi near 30°.

The impact of choice of phi angle is also influenced by the fact thatC_(P) is zero near phi=22° for a round sensor and trends approximatelylinearly toward −1.5 ppm/psi at phi =30° . Since C_(P) is in thedenominator of Equation 14, the effect of the phi angle on thecoefficient in Equation 14 is dramatic. FIG. 6 shows the coefficient inEquation 14 vs. phi angle. It is apparent from FIG. 6 that a roundsensor can be used for a dual-mode pressure transducer for phi anglesgreater than about 25°, where the 10× criteria is approximatelysatisfied. If one is more aggressive and chooses 5× for the criteria,the lower end of the range of usable phi angles falls to 24° . Althoughthe curve in FIG. 6 is theoretical, the experimental point obtained inthis work and shown in FIG. 6 supports the theoretical results.

The use of the sum for f_(C) and f_(B) is a result of concentratingattention on the phi angle range in FIG. 5. Since C_(P) passes throughzero around phi of 22°, at lower phi angles one may use the differencef_(B)−f_(C) to reduce the error arising in transient conditions. Here,using the difference, the appropriate equation relating the pressureerror to H is given by

$\begin{matrix}{{\Delta\; P} = {\frac{C_{T}}{C_{P}}\frac{\left( {{f_{B\; 1}B_{P}} + {f_{C\; 1}C_{P}}} \right)}{\left( {{f_{B\; 1}B_{T}} + {f_{C\; 1}C_{T}}} \right)}{H.}}} & (15)\end{matrix}$

The form of Equation 14 may be maintained by dividing both numerator anddenominator of Equation 15 by −1:

$\begin{matrix}{{\Delta\; P} = {\frac{C_{T}}{C_{P}}\frac{\left( {{f_{C\; 1}C_{P}} + {f_{B\; 1}B_{P}}} \right)}{\left( {{f_{C\; 1}C_{T}} + {f_{B\; 1}B_{T}}} \right)}{H.}}} & (16)\end{matrix}$

The most general form of the present disclosure is based on the factthat once one has the values for f_(C) and f_(B), one is free to performalmost any desired calculation. Thus, we may use f_(C)+K*f_(B) tocompute T, where K is a scalar number. The equations for this case maybe easily derived. Substituting K*f_(B) and K*f_(B1) for f_(B) andf_(B1), respectively, in Equation 11, the following equation forcomputing a change in T can be written as

$\begin{matrix}{{\Delta\; T} = {\frac{\left( {f_{C} + {Kf}_{B} - f_{C\; 1} - {Kf}_{B\; 1}} \right)}{\left( {{f_{C\; 1}C_{T}} + {{Kf}_{B\; 1}B_{T}}} \right)}.}} & (17)\end{matrix}$

If the same substitutions are made into Equation 13, the error in T dueto the presence of H may be found:

$\begin{matrix}{{\Delta\; T} = {\frac{\left( {{f_{C\; 1}C_{P}} + {{Kf}_{B\; 1}B_{P}}} \right)}{\left( {{f_{C\; 1}C_{T}} + {{Kf}_{B\; 1}B_{T}}} \right)}{H.}}} & (18)\end{matrix}$

Equation 18 may be used in Equation 8 to arrive at the most general caseof the present disclosure:

$\begin{matrix}{{\Delta\; P} = {\frac{C_{T}}{C_{P}}\frac{\left( {{f_{C\; 1}C_{P}} + {{Kf}_{B\; 1}B_{P}}} \right)}{\left( {{f_{C\; 1}C_{T}} + {{Kf}_{B\; 1}B_{T}}} \right)}{H.}}} & (19)\end{matrix}$

Note that Equation 19 becomes Equation 14 when K=1, and becomes Equation16 when K=−1. However, K may be adjusted to minimize the termf_(C1)*C_(P)+K*f_(B1)*B_(P) in Equation 19. This has been done for somecalibration data of the sensor used for the experimental point in FIG.5. The result is shown in FIG. 7, where a K of 0.606 was found to reducethe numerator of Equation 19 over the temperature range employed topractically zero. The fact that there is some small T dependence in FIG.7 arises because f_(C) and f_(B) are not perfectly linear with P and T.Since it has already been shown that the value of 0.33 Hz/psi forf_(C)+f_(B) is adequate for actual use, the greatly reduced value inFIG. 7 will provide even superior performance.

Thus, one significant benefit of this disclosure is that by properchoice of the combination f_(C)+K*f_(B), one can now choose the anglephi for the crystallographic orientation of the sensor for otherreasons. One option is to choose a phi angle far from 22° to obtain alarge pressure sensitivity of f_(C). Another important consideration isthat both the C-mode and B-mode must be free of significant activitydips. Yet another consideration is that the resistance of the two modeschanges greatly with phi, so, depending on the circuits to be used inconjunction with the transducer, it may be desirable to adjust phiappropriately.

It should be noted that the use of the deep flats on the transducer bodyas disclosed in the '340 patent might, if desired, be used to improvethe present disclosure in terms of reducing f_(C)+f_(B) over thatobtained from a round-bodied unit. We can understand this by looking atFIG. 5. With judicious choice of the orientation of the flats, thestresses in the resonator become non-uniform. This can cause the curvefor the B-mode to move downward to lower positive values as the Psensitivity decreases, and lower the curve for the C-mode toward largermagnitude, but negative, values for the C-mode as the P sensitivityincreases. This reduces the sum f_(C)+f_(B) over the value for a roundunit. However, the use of flats of any significant depth to createnon-uniform stress distributions in the resonator, and in the end caps,may unfortunately increase the potential for twinning, or cracking. Inaddition, the use of flats may be unnecessary with the presentdisclosure because the function to be provided by the flats can beeffected with f_(C)+K*f_(B).

It should be emphasized that conventional quartz transducer constructionpractices utilize small exterior flats for alignment purposes duringassembly, but such flats are sufficiently small to not cause anyappreciable non-uniform stress in the resonator and, accordingly, theterm “flat” as applied to quartz transducer structures means andincludes a flat or flats of sufficient magnitude to induce non-uniformstress in a resonator of such transducer structures under appliedexterior pressure. For example, a transducer body in accordance with atleast one embodiment of the present disclosure may include two largeflats and two smaller flats, each being offset about 90° about the bodyof the transducer as shown in FIG. 2. Such a configuration may aid inthe assembly of the transducer body by helping to ensure that the endcaps and resonator are assembled in the correct orientation.Accordingly, the term “substantially cylindrical” as used herein withregard to quartz transducer structures means and includes structuresdevoid of a flat or flats of sufficient magnitude to induce non-uniformstress in a resonator of such a quartz transducer structure. Forexample, a substantially cylindrical transducer body may include one ormore flats to aid in the assembly of the transducer body as discussedabove.

FIG. 8 is a schematic diagram of a circuit 100 suitable for use withembodiments of a quartz crystal pressure transducer according toembodiments of the present disclosure. As shown in FIG. 8, the circuit100 includes a first oscillator 102 driven by a first amplifier 104 fordriving a reference crystal (e.g., one of resonators 1 a, 1 b, 1 c(FIGS. 1A-1C)) at a selected frequency (e.g., about 7.2 MHz). Thecircuit 100 includes one more oscillators (e.g., oscillator 106 drivenby amplifier 108) for driving another crystal (e.g., one of resonators 1a, 1 b, 1 c) that acts as a dual-mode sensor. For example, theoscillator 106 may drive the dual-mode sensor crystal at two differentfrequencies (e.g., a C-mode of about 7.24 MHz and a B-mode of about 7.8MHz) to provide both pressure and temperature measurements from a singlecrystal. In other embodiments, two oscillators may be utilized to drivethe single crystal to provide both pressure and temperature measurementsfrom the single crystal. A frequency signal from the reference crystalmay be sent to a processor 110 (e.g., a microcomputer) for furtherprocessing, if desired, and that may be outputted to a reference outputF_(REF).

One or more frequency signals from the dual-mode sensor crystal (e.g.,two frequency signals created by the oscillator 106 driving the crystalat two different frequencies) may be may be sent to the processor 110for further processing, if desired, and for use in the equations fortemperature and pressure as set forth above. The results of thosecalculations may be outputted to output F_(TEMP) and output F_(REF).

In contrast with the state of the art as exemplified by Schodowsky andU.S. Pat. No. 4,545,638 to EerNisse and Ward, embodiments of the presentdisclosure do not employ the use of harmonically related vibrationalmodes that require inclusion of the fundamental mode to obtain therequired temperature sensitivity and, consequently, avoid therequirement of a relatively large resonator bore diameter and theassociate disadvantages indicated above.

For example, in a practical implementation of an embodiment of thepresent disclosure, any harmonic higher than the fundamental is aboutthe same mode shape and, therefore, usable. Consequently, embodiments ofthe present disclosure may employ the 3^(rd) harmonic of both the B- andC-modes, or a 3^(rd) of one of the B-mode and the C-mode and a 5^(th) ofthe other, for temperature calculation and compensation purposes.

While the disclosure may be susceptible to various modifications andalternative forms, specific embodiments have been shown by way ofexample in the drawings and have been described in detail herein.However, it should be understood that the invention is not intended tobe limited to the particular forms disclosed. Rather, the invention isto cover all modifications, equivalents, and alternatives falling withinthe spirit and scope of the invention as defined by the followingappended claims.

What is claimed is:
 1. A method of using a quartz crystal structure of apressure transducer, the method comprising: stimulating, under transienttemperature conditions, a resonator of the quartz crystal structureunder external pressure applied to the quartz crystal structure toprovide signal inputs to an electronics assembly associated with theresonator, the signal inputs comprising a first signal input from anon-fundamental B-mode resonant frequency and a second signal input froma non-fundamental C-mode resonant frequency; and using a combination ofthe first signal input and the second signal input to compensate apressure determined from the second signal input of the non-fundamentalC-mode resonant frequency signal input, comprising: receiving with theelectronics assembly from the resonator the first signal input of thenon-fundamental B-mode resonant frequency that is primarily dependent ontemperature; receiving with the electronics assembly from the resonatorthe second signal input of the non-fundamental C-mode resonant frequencythat is primarily dependent on pressure; calculating with theelectronics assembly substantially pressure-independent temperatureunder the transient temperature conditions using a combination of thefirst signal input and the second signal input and compensating with theelectronics assembly the pressure determined from the signal input ofthe non-fundamental C-mode resonant frequency with data from thecalculation of the substantially pressure-independent temperature. 2.The method of claim 1, wherein using a combination of the first signalinput and the second signal input comprises using a sum of the firstsignal input and the second signal input.
 3. The method of claim 1,wherein the non-fundamental B-mode resonant frequency and thenon-fundamental C-mode resonant frequency is the 3 ^(rd) harmonic ofeach mode.
 4. The method of claim 1, further comprising determining achange in temperature with the following equation: $\begin{matrix}{{\Delta\; T} = {\frac{\left( {f_{C} + {Kf}_{B} - f_{C\; 1} - {Kf}_{B\; 1}} \right)}{\left( {{f_{C\; 1}C_{T}} + {{Kf}_{B\; 1}B_{T}}} \right)}.}} & \;\end{matrix}$
 5. The method of claim 1, further comprising determining achange in pressure with the following equation:${\Delta\; P} = {\frac{C_{T}}{C_{P}}\frac{\left( {{f_{C\; 1}C_{P}} + {{Kf}_{B\; 1}B_{P}}} \right)}{\left( {{f_{C\; 1}C_{T}} + {{Kf}_{B\; 1}B_{T}}} \right)}{H.}}$6. The method of claim 5, further comprising selecting a value of theconstant K in the equation defining the change in pressure to minimizetemperature dependence of the equation defining the change in pressure.7. The method of claim 1, wherein stimulating the resonator comprisesdriving the resonator at least two distinct frequencies with at leastone oscillator.
 8. The method of claim 1, wherein stimulating theresonator comprises driving the resonator comprising a single crystalwith at least one oscillator at a first frequency and a second frequencythat is greater than the first frequency to provide both pressure andtemperature measurements from the single crystal.
 9. The method of claim1, further comprising determining both pressure and temperaturemeasurements from the resonator comprising a single crystal.
 10. Amethod of using a quartz crystal structure of a pressure transducer, themethod comprising: stimulating a resonator of the quartz crystalstructure of the pressure transducer with an electronics assemblyassociated with the pressure transducer under external pressure appliedto the quartz crystal structure; receiving with the electronics assemblyfrom the resonator a non-fundamental B-mode resonant frequency signalinput primarily dependent on temperature and a non-fundamental C-moderesonant frequency signal input from the resonator primarily dependenton pressure; and determining both a pressure measurement and atemperature measurement from the resonator using both of thenon-fundamental B-mode resonant frequency signal input and thenon-fundamental C-mode resonant frequency signal input.
 11. The methodof claim 10, wherein determining both the pressure measurement and thetemperature measurement from the resonator comprises determining thetemperature measurement using both of the non-fundamental B-moderesonant frequency signal input and the non-fundamental C-mode resonantfrequency signal input.
 12. The method of claim 11, wherein determiningboth the pressure measurement and the temperature measurement from theresonator further comprises compensating the pressure measurement usingdata from the determining of the temperature measurement using both ofthe non-fundamental B-mode resonant frequency signal input and thenon-fundamental C-mode resonant frequency signal input.
 13. The methodof claim 10, wherein determining both the pressure measurement and thetemperature measurement from the resonator comprises determining thetemperature measurement under transient temperature conditions usingboth of the non-fundamental B-mode resonant frequency signal input andthe non-fundamental C-mode resonant frequency signal input.
 14. Themethod of claim 10, further comprising utilizing a 3^(rd) harmonic ofeach of the non-fundamental B-mode resonant frequency signal input andthe non-fundamental C-mode resonant frequency signal input.
 15. Themethod of claim 10, further comprising utilizing a 3^(rd) harmonic ofone of the non-fundamental B-mode resonant frequency signal input andthe non-fundamental C-mode resonant frequency signal input and a 5^(th)harmonic of the other one of the non-fundamental B-mode resonantfrequency signal input and the non-fundamental C-mode resonant frequencysignal input.
 16. A method of using a quartz crystal structure of apressure transducer, the method comprising: stimulating, under transientconditions, a resonator of the quartz crystal structure of the pressuretransducer with an electronics assembly associated with the pressuretransducer under external pressure applied to the quartz crystalstructure; receiving with the electronics assembly a non-fundamentalB-mode resonant frequency signal input and a non-fundamental C-moderesonant frequency signal input from the resonator; and determiningtemperature-compensated pressure from the non-fundamental B-moderesonant frequency signal input and the non-fundamental C-mode resonantfrequency signal input.
 17. The method of claim 15, wherein determiningthe temperature-compensated pressure comprises receiving with theelectronics assembly the non-fundamental B-mode resonant frequencysignal input that is primarily dependent on temperature.
 18. The methodof claim 16, wherein determining the temperature-compensated pressurefurther comprises receiving with the electronics assembly thenon-fundamental C-mode resonant frequency signal input that is primarilydependent on pressure.
 19. The method of claim 16, wherein determiningthe temperature-compensated pressure comprises calculating with theelectronics assembly substantially pressure-independent temperatureunder the transient conditions using a combination of thenon-fundamental B-mode resonant frequency signal input and thenon-fundamental C-mode resonant frequency signal input.
 20. The methodof claim 19, wherein determining the temperature-compensated pressurefurther comprises compensating for transient temperature conditions inthe pressure determined from the non-fundamental C-mode resonantfrequency signal input using the substantially pressure-independenttemperature.